Metamath Proof Explorer

Theorem mulsproplem8

Description: Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 5-Mar-2025)

Ref Expression
Hypotheses mulsproplem.1 
|- ( ph -> A. a e. No A. b e. No A. c e. No A. d e. No A. e e. No A. f e. No ( ( ( ( bday ` a ) +no ( bday ` b ) ) u. ( ( ( ( bday ` c ) +no ( bday ` e ) ) u. ( ( bday ` d ) +no ( bday ` f ) ) ) u. ( ( ( bday ` c ) +no ( bday ` f ) ) u. ( ( bday ` d ) +no ( bday ` e ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) -> ( ( a x.s b ) e. No /\ ( ( c  ( ( c x.s f ) -s ( c x.s e ) )
mulsproplem8.1 
|- ( ph -> A e. No )
mulsproplem8.2 
|- ( ph -> B e. No )
mulsproplem8.3 
|- ( ph -> R e. ( _Right ` A ) )
mulsproplem8.4 
|- ( ph -> S e. ( _Right ` B ) )
mulsproplem8.5 
|- ( ph -> V e. ( _Right ` A ) )
mulsproplem8.6 
|- ( ph -> W e. ( _Left ` B ) )
Assertion mulsproplem8 
|- ( ph -> ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )

Proof

Step Hyp Ref Expression
1 mulsproplem.1 
 |-  ( ph -> A. a e. No A. b e. No A. c e. No A. d e. No A. e e. No A. f e. No ( ( ( ( bday ` a ) +no ( bday ` b ) ) u. ( ( ( ( bday ` c ) +no ( bday ` e ) ) u. ( ( bday ` d ) +no ( bday ` f ) ) ) u. ( ( ( bday ` c ) +no ( bday ` f ) ) u. ( ( bday ` d ) +no ( bday ` e ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) -> ( ( a x.s b ) e. No /\ ( ( c  ( ( c x.s f ) -s ( c x.s e ) )
2 mulsproplem8.1 
 |-  ( ph -> A e. No )
3 mulsproplem8.2 
 |-  ( ph -> B e. No )
4 mulsproplem8.3 
 |-  ( ph -> R e. ( _Right ` A ) )
5 mulsproplem8.4 
 |-  ( ph -> S e. ( _Right ` B ) )
6 mulsproplem8.5 
 |-  ( ph -> V e. ( _Right ` A ) )
7 mulsproplem8.6 
 |-  ( ph -> W e. ( _Left ` B ) )
8 4 rightnod 
 |-  ( ph -> R e. No )
9 6 rightnod 
 |-  ( ph -> V e. No )
10 ltslin 
 |-  ( ( R e. No /\ V e. No ) -> ( R
11 8 9 10 syl2anc 
 |-  ( ph -> ( R
12 4 rightoldd 
 |-  ( ph -> R e. ( _Old ` ( bday ` A ) ) )
13 1 12 3 mulsproplem2 
 |-  ( ph -> ( R x.s B ) e. No )
14 5 rightoldd 
 |-  ( ph -> S e. ( _Old ` ( bday ` B ) ) )
15 1 2 14 mulsproplem3 
 |-  ( ph -> ( A x.s S ) e. No )
16 13 15 addscld 
 |-  ( ph -> ( ( R x.s B ) +s ( A x.s S ) ) e. No )
17 1 12 14 mulsproplem4 
 |-  ( ph -> ( R x.s S ) e. No )
18 16 17 subscld 
 |-  ( ph -> ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) ) e. No )
19 18 adantr 
 |-  ( ( ph /\ R  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) ) e. No )
20 7 leftoldd 
 |-  ( ph -> W e. ( _Old ` ( bday ` B ) ) )
21 1 2 20 mulsproplem3 
 |-  ( ph -> ( A x.s W ) e. No )
22 13 21 addscld 
 |-  ( ph -> ( ( R x.s B ) +s ( A x.s W ) ) e. No )
23 1 12 20 mulsproplem4 
 |-  ( ph -> ( R x.s W ) e. No )
24 22 23 subscld 
 |-  ( ph -> ( ( ( R x.s B ) +s ( A x.s W ) ) -s ( R x.s W ) ) e. No )
25 24 adantr 
 |-  ( ( ph /\ R  ( ( ( R x.s B ) +s ( A x.s W ) ) -s ( R x.s W ) ) e. No )
26 6 rightoldd 
 |-  ( ph -> V e. ( _Old ` ( bday ` A ) ) )
27 1 26 3 mulsproplem2 
 |-  ( ph -> ( V x.s B ) e. No )
28 27 21 addscld 
 |-  ( ph -> ( ( V x.s B ) +s ( A x.s W ) ) e. No )
29 1 26 20 mulsproplem4 
 |-  ( ph -> ( V x.s W ) e. No )
30 28 29 subscld 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) e. No )
31 30 adantr 
 |-  ( ( ph /\ R  ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) e. No )
32 sltsright 
 |-  ( A e. No -> { A } <
33 2 32 syl 
 |-  ( ph -> { A } <
34 snidg 
 |-  ( A e. No -> A e. { A } )
35 2 34 syl 
 |-  ( ph -> A e. { A } )
36 33 35 4 sltssepcd 
 |-  ( ph -> A
37 lltr 
 |-  ( _Left ` B ) <
38 37 a1i 
 |-  ( ph -> ( _Left ` B ) <
39 38 7 5 sltssepcd 
 |-  ( ph -> W
40 0no 
 |-  0s e. No
41 40 a1i 
 |-  ( ph -> 0s e. No )
42 7 leftnod 
 |-  ( ph -> W e. No )
43 5 rightnod 
 |-  ( ph -> S e. No )
44 bday0 
 |-  ( bday ` 0s ) = (/)
45 44 44 oveq12i 
 |-  ( ( bday ` 0s ) +no ( bday ` 0s ) ) = ( (/) +no (/) )
46 0elon 
 |-  (/) e. On
47 naddrid 
 |-  ( (/) e. On -> ( (/) +no (/) ) = (/) )
48 46 47 ax-mp 
 |-  ( (/) +no (/) ) = (/)
49 45 48 eqtri 
 |-  ( ( bday ` 0s ) +no ( bday ` 0s ) ) = (/)
50 49 uneq1i 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) ) = ( (/) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) )
51 0un 
 |-  ( (/) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) )
52 50 51 eqtri 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) )
53 oldbdayim 
 |-  ( W e. ( _Old ` ( bday ` B ) ) -> ( bday ` W ) e. ( bday ` B ) )
54 20 53 syl 
 |-  ( ph -> ( bday ` W ) e. ( bday ` B ) )
55 bdayon 
 |-  ( bday ` W ) e. On
56 bdayon 
 |-  ( bday ` B ) e. On
57 bdayon 
 |-  ( bday ` A ) e. On
58 naddel2 
 |-  ( ( ( bday ` W ) e. On /\ ( bday ` B ) e. On /\ ( bday ` A ) e. On ) -> ( ( bday ` W ) e. ( bday ` B ) <-> ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
59 55 56 57 58 mp3an 
 |-  ( ( bday ` W ) e. ( bday ` B ) <-> ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
60 54 59 sylib 
 |-  ( ph -> ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
61 oldbdayim 
 |-  ( R e. ( _Old ` ( bday ` A ) ) -> ( bday ` R ) e. ( bday ` A ) )
62 12 61 syl 
 |-  ( ph -> ( bday ` R ) e. ( bday ` A ) )
63 oldbdayim 
 |-  ( S e. ( _Old ` ( bday ` B ) ) -> ( bday ` S ) e. ( bday ` B ) )
64 14 63 syl 
 |-  ( ph -> ( bday ` S ) e. ( bday ` B ) )
65 naddel12 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( ( bday ` R ) e. ( bday ` A ) /\ ( bday ` S ) e. ( bday ` B ) ) -> ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
66 57 56 65 mp2an 
 |-  ( ( ( bday ` R ) e. ( bday ` A ) /\ ( bday ` S ) e. ( bday ` B ) ) -> ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
67 62 64 66 syl2anc 
 |-  ( ph -> ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
68 60 67 jca 
 |-  ( ph -> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
69 bdayon 
 |-  ( bday ` S ) e. On
70 naddel2 
 |-  ( ( ( bday ` S ) e. On /\ ( bday ` B ) e. On /\ ( bday ` A ) e. On ) -> ( ( bday ` S ) e. ( bday ` B ) <-> ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
71 69 56 57 70 mp3an 
 |-  ( ( bday ` S ) e. ( bday ` B ) <-> ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
72 64 71 sylib 
 |-  ( ph -> ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
73 naddel12 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( ( bday ` R ) e. ( bday ` A ) /\ ( bday ` W ) e. ( bday ` B ) ) -> ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
74 57 56 73 mp2an 
 |-  ( ( ( bday ` R ) e. ( bday ` A ) /\ ( bday ` W ) e. ( bday ` B ) ) -> ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
75 62 54 74 syl2anc 
 |-  ( ph -> ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
76 72 75 jca 
 |-  ( ph -> ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
77 naddcl 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` W ) e. On ) -> ( ( bday ` A ) +no ( bday ` W ) ) e. On )
78 57 55 77 mp2an 
 |-  ( ( bday ` A ) +no ( bday ` W ) ) e. On
79 bdayon 
 |-  ( bday ` R ) e. On
80 naddcl 
 |-  ( ( ( bday ` R ) e. On /\ ( bday ` S ) e. On ) -> ( ( bday ` R ) +no ( bday ` S ) ) e. On )
81 79 69 80 mp2an 
 |-  ( ( bday ` R ) +no ( bday ` S ) ) e. On
82 78 81 onun2i 
 |-  ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. On
83 naddcl 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` S ) e. On ) -> ( ( bday ` A ) +no ( bday ` S ) ) e. On )
84 57 69 83 mp2an 
 |-  ( ( bday ` A ) +no ( bday ` S ) ) e. On
85 naddcl 
 |-  ( ( ( bday ` R ) e. On /\ ( bday ` W ) e. On ) -> ( ( bday ` R ) +no ( bday ` W ) ) e. On )
86 79 55 85 mp2an 
 |-  ( ( bday ` R ) +no ( bday ` W ) ) e. On
87 84 86 onun2i 
 |-  ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) e. On
88 naddcl 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` A ) +no ( bday ` B ) ) e. On )
89 57 56 88 mp2an 
 |-  ( ( bday ` A ) +no ( bday ` B ) ) e. On
90 onunel 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. On /\ ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
91 82 87 89 90 mp3an 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
92 onunel 
 |-  ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. On /\ ( ( bday ` R ) +no ( bday ` S ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
93 78 81 89 92 mp3an 
 |-  ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
94 onunel 
 |-  ( ( ( ( bday ` A ) +no ( bday ` S ) ) e. On /\ ( ( bday ` R ) +no ( bday ` W ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
95 84 86 89 94 mp3an 
 |-  ( ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
96 93 95 anbi12i 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
97 91 96 bitri 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
98 68 76 97 sylanbrc 
 |-  ( ph -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
99 elun1 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
100 98 99 syl 
 |-  ( ph -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
101 52 100 eqeltrid 
 |-  ( ph -> ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` W ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
102 1 41 41 2 8 42 43 101 mulsproplem1 
 |-  ( ph -> ( ( 0s x.s 0s ) e. No /\ ( ( A  ( ( A x.s S ) -s ( A x.s W ) )
103 102 simprd 
 |-  ( ph -> ( ( A  ( ( A x.s S ) -s ( A x.s W ) )
104 36 39 103 mp2and 
 |-  ( ph -> ( ( A x.s S ) -s ( A x.s W ) )
105 15 17 21 23 ltsubsubsbd 
 |-  ( ph -> ( ( ( A x.s S ) -s ( A x.s W ) )  ( ( A x.s S ) -s ( R x.s S ) )
106 15 17 subscld 
 |-  ( ph -> ( ( A x.s S ) -s ( R x.s S ) ) e. No )
107 21 23 subscld 
 |-  ( ph -> ( ( A x.s W ) -s ( R x.s W ) ) e. No )
108 106 107 13 ltadds2d 
 |-  ( ph -> ( ( ( A x.s S ) -s ( R x.s S ) )  ( ( R x.s B ) +s ( ( A x.s S ) -s ( R x.s S ) ) )
109 105 108 bitrd 
 |-  ( ph -> ( ( ( A x.s S ) -s ( A x.s W ) )  ( ( R x.s B ) +s ( ( A x.s S ) -s ( R x.s S ) ) )
110 104 109 mpbid 
 |-  ( ph -> ( ( R x.s B ) +s ( ( A x.s S ) -s ( R x.s S ) ) )
111 13 15 17 addsubsassd 
 |-  ( ph -> ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) ) = ( ( R x.s B ) +s ( ( A x.s S ) -s ( R x.s S ) ) ) )
112 13 21 23 addsubsassd 
 |-  ( ph -> ( ( ( R x.s B ) +s ( A x.s W ) ) -s ( R x.s W ) ) = ( ( R x.s B ) +s ( ( A x.s W ) -s ( R x.s W ) ) ) )
113 110 111 112 3brtr4d 
 |-  ( ph -> ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
114 113 adantr 
 |-  ( ( ph /\ R  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
115 sltsleft 
 |-  ( B e. No -> ( _Left ` B ) <
116 3 115 syl 
 |-  ( ph -> ( _Left ` B ) <
117 snidg 
 |-  ( B e. No -> B e. { B } )
118 3 117 syl 
 |-  ( ph -> B e. { B } )
119 116 7 118 sltssepcd 
 |-  ( ph -> W
120 49 uneq1i 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( (/) u. ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) )
121 0un 
 |-  ( (/) u. ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) )
122 120 121 eqtri 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) )
123 oldbdayim 
 |-  ( V e. ( _Old ` ( bday ` A ) ) -> ( bday ` V ) e. ( bday ` A ) )
124 26 123 syl 
 |-  ( ph -> ( bday ` V ) e. ( bday ` A ) )
125 bdayon 
 |-  ( bday ` V ) e. On
126 naddel1 
 |-  ( ( ( bday ` V ) e. On /\ ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` V ) e. ( bday ` A ) <-> ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
127 125 57 56 126 mp3an 
 |-  ( ( bday ` V ) e. ( bday ` A ) <-> ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
128 124 127 sylib 
 |-  ( ph -> ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
129 75 128 jca 
 |-  ( ph -> ( ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
130 naddel1 
 |-  ( ( ( bday ` R ) e. On /\ ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` R ) e. ( bday ` A ) <-> ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
131 79 57 56 130 mp3an 
 |-  ( ( bday ` R ) e. ( bday ` A ) <-> ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
132 62 131 sylib 
 |-  ( ph -> ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
133 naddel12 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( ( bday ` V ) e. ( bday ` A ) /\ ( bday ` W ) e. ( bday ` B ) ) -> ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
134 57 56 133 mp2an 
 |-  ( ( ( bday ` V ) e. ( bday ` A ) /\ ( bday ` W ) e. ( bday ` B ) ) -> ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
135 124 54 134 syl2anc 
 |-  ( ph -> ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
136 132 135 jca 
 |-  ( ph -> ( ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
137 naddcl 
 |-  ( ( ( bday ` V ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` V ) +no ( bday ` B ) ) e. On )
138 125 56 137 mp2an 
 |-  ( ( bday ` V ) +no ( bday ` B ) ) e. On
139 86 138 onun2i 
 |-  ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. On
140 naddcl 
 |-  ( ( ( bday ` R ) e. On /\ ( bday ` B ) e. On ) -> ( ( bday ` R ) +no ( bday ` B ) ) e. On )
141 79 56 140 mp2an 
 |-  ( ( bday ` R ) +no ( bday ` B ) ) e. On
142 naddcl 
 |-  ( ( ( bday ` V ) e. On /\ ( bday ` W ) e. On ) -> ( ( bday ` V ) +no ( bday ` W ) ) e. On )
143 125 55 142 mp2an 
 |-  ( ( bday ` V ) +no ( bday ` W ) ) e. On
144 141 143 onun2i 
 |-  ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. On
145 onunel 
 |-  ( ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. On /\ ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
146 139 144 89 145 mp3an 
 |-  ( ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
147 onunel 
 |-  ( ( ( ( bday ` R ) +no ( bday ` W ) ) e. On /\ ( ( bday ` V ) +no ( bday ` B ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
148 86 138 89 147 mp3an 
 |-  ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
149 onunel 
 |-  ( ( ( ( bday ` R ) +no ( bday ` B ) ) e. On /\ ( ( bday ` V ) +no ( bday ` W ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
150 141 143 89 149 mp3an 
 |-  ( ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
151 148 150 anbi12i 
 |-  ( ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) <-> ( ( ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
152 146 151 bitri 
 |-  ( ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` R ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
153 129 136 152 sylanbrc 
 |-  ( ph -> ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
154 elun1 
 |-  ( ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) -> ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
155 153 154 syl 
 |-  ( ph -> ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
156 122 155 eqeltrid 
 |-  ( ph -> ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` R ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` B ) ) ) u. ( ( ( bday ` R ) +no ( bday ` B ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
157 1 41 41 8 9 42 3 156 mulsproplem1 
 |-  ( ph -> ( ( 0s x.s 0s ) e. No /\ ( ( R  ( ( R x.s B ) -s ( R x.s W ) )
158 157 simprd 
 |-  ( ph -> ( ( R  ( ( R x.s B ) -s ( R x.s W ) )
159 119 158 mpan2d 
 |-  ( ph -> ( R  ( ( R x.s B ) -s ( R x.s W ) )
160 159 imp 
 |-  ( ( ph /\ R  ( ( R x.s B ) -s ( R x.s W ) )
161 13 23 subscld 
 |-  ( ph -> ( ( R x.s B ) -s ( R x.s W ) ) e. No )
162 27 29 subscld 
 |-  ( ph -> ( ( V x.s B ) -s ( V x.s W ) ) e. No )
163 161 162 21 ltadds1d 
 |-  ( ph -> ( ( ( R x.s B ) -s ( R x.s W ) )  ( ( ( R x.s B ) -s ( R x.s W ) ) +s ( A x.s W ) )
164 163 adantr 
 |-  ( ( ph /\ R  ( ( ( R x.s B ) -s ( R x.s W ) )  ( ( ( R x.s B ) -s ( R x.s W ) ) +s ( A x.s W ) )
165 160 164 mpbid 
 |-  ( ( ph /\ R  ( ( ( R x.s B ) -s ( R x.s W ) ) +s ( A x.s W ) )
166 13 21 23 addsubsd 
 |-  ( ph -> ( ( ( R x.s B ) +s ( A x.s W ) ) -s ( R x.s W ) ) = ( ( ( R x.s B ) -s ( R x.s W ) ) +s ( A x.s W ) ) )
167 166 adantr 
 |-  ( ( ph /\ R  ( ( ( R x.s B ) +s ( A x.s W ) ) -s ( R x.s W ) ) = ( ( ( R x.s B ) -s ( R x.s W ) ) +s ( A x.s W ) ) )
168 27 21 29 addsubsd 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) = ( ( ( V x.s B ) -s ( V x.s W ) ) +s ( A x.s W ) ) )
169 168 adantr 
 |-  ( ( ph /\ R  ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) = ( ( ( V x.s B ) -s ( V x.s W ) ) +s ( A x.s W ) ) )
170 165 167 169 3brtr4d 
 |-  ( ( ph /\ R  ( ( ( R x.s B ) +s ( A x.s W ) ) -s ( R x.s W ) )
171 19 25 31 114 170 ltstrd 
 |-  ( ( ph /\ R  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
172 171 ex 
 |-  ( ph -> ( R  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
173 33 35 6 sltssepcd 
 |-  ( ph -> A
174 49 uneq1i 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( (/) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) )
175 0un 
 |-  ( (/) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) )
176 174 175 eqtri 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) = ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) )
177 naddel12 
 |-  ( ( ( bday ` A ) e. On /\ ( bday ` B ) e. On ) -> ( ( ( bday ` V ) e. ( bday ` A ) /\ ( bday ` S ) e. ( bday ` B ) ) -> ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
178 57 56 177 mp2an 
 |-  ( ( ( bday ` V ) e. ( bday ` A ) /\ ( bday ` S ) e. ( bday ` B ) ) -> ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
179 124 64 178 syl2anc 
 |-  ( ph -> ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
180 60 179 jca 
 |-  ( ph -> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
181 72 135 jca 
 |-  ( ph -> ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
182 naddcl 
 |-  ( ( ( bday ` V ) e. On /\ ( bday ` S ) e. On ) -> ( ( bday ` V ) +no ( bday ` S ) ) e. On )
183 125 69 182 mp2an 
 |-  ( ( bday ` V ) +no ( bday ` S ) ) e. On
184 78 183 onun2i 
 |-  ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) e. On
185 84 143 onun2i 
 |-  ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. On
186 onunel 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) e. On /\ ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
187 184 185 89 186 mp3an 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
188 onunel 
 |-  ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. On /\ ( ( bday ` V ) +no ( bday ` S ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
189 78 183 89 188 mp3an 
 |-  ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
190 onunel 
 |-  ( ( ( ( bday ` A ) +no ( bday ` S ) ) e. On /\ ( ( bday ` V ) +no ( bday ` W ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
191 84 143 89 190 mp3an 
 |-  ( ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
192 189 191 anbi12i 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
193 187 192 bitri 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` A ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` A ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` V ) +no ( bday ` W ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
194 180 181 193 sylanbrc 
 |-  ( ph -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
195 elun1 
 |-  ( ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
196 194 195 syl 
 |-  ( ph -> ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
197 176 196 eqeltrid 
 |-  ( ph -> ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` A ) +no ( bday ` W ) ) u. ( ( bday ` V ) +no ( bday ` S ) ) ) u. ( ( ( bday ` A ) +no ( bday ` S ) ) u. ( ( bday ` V ) +no ( bday ` W ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
198 1 41 41 2 9 42 43 197 mulsproplem1 
 |-  ( ph -> ( ( 0s x.s 0s ) e. No /\ ( ( A  ( ( A x.s S ) -s ( A x.s W ) )
199 198 simprd 
 |-  ( ph -> ( ( A  ( ( A x.s S ) -s ( A x.s W ) )
200 173 39 199 mp2and 
 |-  ( ph -> ( ( A x.s S ) -s ( A x.s W ) )
201 1 26 14 mulsproplem4 
 |-  ( ph -> ( V x.s S ) e. No )
202 15 201 21 29 ltsubsubsbd 
 |-  ( ph -> ( ( ( A x.s S ) -s ( A x.s W ) )  ( ( A x.s S ) -s ( V x.s S ) )
203 15 201 subscld 
 |-  ( ph -> ( ( A x.s S ) -s ( V x.s S ) ) e. No )
204 21 29 subscld 
 |-  ( ph -> ( ( A x.s W ) -s ( V x.s W ) ) e. No )
205 203 204 27 ltadds2d 
 |-  ( ph -> ( ( ( A x.s S ) -s ( V x.s S ) )  ( ( V x.s B ) +s ( ( A x.s S ) -s ( V x.s S ) ) )
206 202 205 bitrd 
 |-  ( ph -> ( ( ( A x.s S ) -s ( A x.s W ) )  ( ( V x.s B ) +s ( ( A x.s S ) -s ( V x.s S ) ) )
207 200 206 mpbid 
 |-  ( ph -> ( ( V x.s B ) +s ( ( A x.s S ) -s ( V x.s S ) ) )
208 27 15 201 addsubsassd 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) ) = ( ( V x.s B ) +s ( ( A x.s S ) -s ( V x.s S ) ) ) )
209 27 21 29 addsubsassd 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) = ( ( V x.s B ) +s ( ( A x.s W ) -s ( V x.s W ) ) ) )
210 207 208 209 3brtr4d 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) )
211 oveq1 
 |-  ( R = V -> ( R x.s B ) = ( V x.s B ) )
212 211 oveq1d 
 |-  ( R = V -> ( ( R x.s B ) +s ( A x.s S ) ) = ( ( V x.s B ) +s ( A x.s S ) ) )
213 oveq1 
 |-  ( R = V -> ( R x.s S ) = ( V x.s S ) )
214 212 213 oveq12d 
 |-  ( R = V -> ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) ) = ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) ) )
215 214 breq1d 
 |-  ( R = V -> ( ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )  ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) )
216 210 215 syl5ibrcom 
 |-  ( ph -> ( R = V -> ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
217 18 adantr 
 |-  ( ( ph /\ V  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) ) e. No )
218 27 15 addscld 
 |-  ( ph -> ( ( V x.s B ) +s ( A x.s S ) ) e. No )
219 218 201 subscld 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) ) e. No )
220 219 adantr 
 |-  ( ( ph /\ V  ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) ) e. No )
221 30 adantr 
 |-  ( ( ph /\ V  ( ( ( V x.s B ) +s ( A x.s W ) ) -s ( V x.s W ) ) e. No )
222 sltsright 
 |-  ( B e. No -> { B } <
223 3 222 syl 
 |-  ( ph -> { B } <
224 223 118 5 sltssepcd 
 |-  ( ph -> B
225 49 uneq1i 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) ) = ( (/) u. ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) )
226 0un 
 |-  ( (/) u. ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) ) = ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) )
227 225 226 eqtri 
 |-  ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) ) = ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) )
228 128 67 jca 
 |-  ( ph -> ( ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
229 179 132 jca 
 |-  ( ph -> ( ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
230 138 81 onun2i 
 |-  ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. On
231 183 141 onun2i 
 |-  ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) e. On
232 onunel 
 |-  ( ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. On /\ ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
233 230 231 89 232 mp3an 
 |-  ( ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
234 onunel 
 |-  ( ( ( ( bday ` V ) +no ( bday ` B ) ) e. On /\ ( ( bday ` R ) +no ( bday ` S ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
235 138 81 89 234 mp3an 
 |-  ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
236 onunel 
 |-  ( ( ( ( bday ` V ) +no ( bday ` S ) ) e. On /\ ( ( bday ` R ) +no ( bday ` B ) ) e. On /\ ( ( bday ` A ) +no ( bday ` B ) ) e. On ) -> ( ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
237 183 141 89 236 mp3an 
 |-  ( ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) )
238 235 237 anbi12i 
 |-  ( ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) <-> ( ( ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
239 233 238 bitri 
 |-  ( ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) <-> ( ( ( ( bday ` V ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) /\ ( ( ( bday ` V ) +no ( bday ` S ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) /\ ( ( bday ` R ) +no ( bday ` B ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) ) ) )
240 228 229 239 sylanbrc 
 |-  ( ph -> ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) )
241 elun1 
 |-  ( ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) e. ( ( bday ` A ) +no ( bday ` B ) ) -> ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
242 240 241 syl 
 |-  ( ph -> ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
243 227 242 eqeltrid 
 |-  ( ph -> ( ( ( bday ` 0s ) +no ( bday ` 0s ) ) u. ( ( ( ( bday ` V ) +no ( bday ` B ) ) u. ( ( bday ` R ) +no ( bday ` S ) ) ) u. ( ( ( bday ` V ) +no ( bday ` S ) ) u. ( ( bday ` R ) +no ( bday ` B ) ) ) ) ) e. ( ( ( bday ` A ) +no ( bday ` B ) ) u. ( ( ( ( bday ` C ) +no ( bday ` E ) ) u. ( ( bday ` D ) +no ( bday ` F ) ) ) u. ( ( ( bday ` C ) +no ( bday ` F ) ) u. ( ( bday ` D ) +no ( bday ` E ) ) ) ) ) )
244 1 41 41 9 8 3 43 243 mulsproplem1 
 |-  ( ph -> ( ( 0s x.s 0s ) e. No /\ ( ( V  ( ( V x.s S ) -s ( V x.s B ) )
245 244 simprd 
 |-  ( ph -> ( ( V  ( ( V x.s S ) -s ( V x.s B ) )
246 224 245 mpan2d 
 |-  ( ph -> ( V  ( ( V x.s S ) -s ( V x.s B ) )
247 246 imp 
 |-  ( ( ph /\ V  ( ( V x.s S ) -s ( V x.s B ) )
248 201 27 17 13 ltsubsubs2bd 
 |-  ( ph -> ( ( ( V x.s S ) -s ( V x.s B ) )  ( ( R x.s B ) -s ( R x.s S ) )
249 13 17 subscld 
 |-  ( ph -> ( ( R x.s B ) -s ( R x.s S ) ) e. No )
250 27 201 subscld 
 |-  ( ph -> ( ( V x.s B ) -s ( V x.s S ) ) e. No )
251 249 250 15 ltadds1d 
 |-  ( ph -> ( ( ( R x.s B ) -s ( R x.s S ) )  ( ( ( R x.s B ) -s ( R x.s S ) ) +s ( A x.s S ) )
252 248 251 bitrd 
 |-  ( ph -> ( ( ( V x.s S ) -s ( V x.s B ) )  ( ( ( R x.s B ) -s ( R x.s S ) ) +s ( A x.s S ) )
253 252 adantr 
 |-  ( ( ph /\ V  ( ( ( V x.s S ) -s ( V x.s B ) )  ( ( ( R x.s B ) -s ( R x.s S ) ) +s ( A x.s S ) )
254 247 253 mpbid 
 |-  ( ( ph /\ V  ( ( ( R x.s B ) -s ( R x.s S ) ) +s ( A x.s S ) )
255 13 15 17 addsubsd 
 |-  ( ph -> ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) ) = ( ( ( R x.s B ) -s ( R x.s S ) ) +s ( A x.s S ) ) )
256 255 adantr 
 |-  ( ( ph /\ V  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) ) = ( ( ( R x.s B ) -s ( R x.s S ) ) +s ( A x.s S ) ) )
257 27 15 201 addsubsd 
 |-  ( ph -> ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) ) = ( ( ( V x.s B ) -s ( V x.s S ) ) +s ( A x.s S ) ) )
258 257 adantr 
 |-  ( ( ph /\ V  ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) ) = ( ( ( V x.s B ) -s ( V x.s S ) ) +s ( A x.s S ) ) )
259 254 256 258 3brtr4d 
 |-  ( ( ph /\ V  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
260 210 adantr 
 |-  ( ( ph /\ V  ( ( ( V x.s B ) +s ( A x.s S ) ) -s ( V x.s S ) )
261 217 220 221 259 260 ltstrd 
 |-  ( ( ph /\ V  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
262 261 ex 
 |-  ( ph -> ( V  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
263 172 216 262 3jaod 
 |-  ( ph -> ( ( R  ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )
264 11 263 mpd 
 |-  ( ph -> ( ( ( R x.s B ) +s ( A x.s S ) ) -s ( R x.s S ) )